# Probability Calculator

The Probability Calculator computes the probability of one event, based on probabilities of other events. It displays the answer, and writes a report that explains how to compute the answer.

• Specify the main goal of the analysis (via the dropdown box).
• Enter required probabilities in the unshaded text boxes (see Hint below).
• Click the Calculate button to create a summary report.
Main goal:
 Hint: To solve this problem, provide input from one of the options listed below.Option 1. P(A'). Option 2. P(B), P(A ∪ B), and P(A ∩ B). Option 3. P( B|A ) and P(A ∩ B). Probability of event A:P( A ) Probability that event A does not occur: P( A' ) Probability of event B:P( B ) Conditional probability of B, given A: P( B|A ) Probability of the union of A and B: P( A ∪ B ) Probability of intersection of A and B: P( A ∩ B )

This calculator handles problems that can be addressed using three basic rules of probability - the subtraction rule, addition rule, and multiplication rule. For problems that require Bayes' rule, use the Bayes' Rule Calculator.

## Summary Report

To create a report, enter data into the Probability Calculator and click the Calculate button.

Instructions: To find the answer to a frequently-asked question, simply click on the question.

### What if I don't understand the notation?

All of the notation used by the Probability Calculator is defined below:

• P( A ):
Probability of event A
• P( A' ):
Probability that event A does not occur
• P( B|A ):
Conditional probability of event B, given event A
• P(A B):
Probability that event A and/or event B occurs. This is also known as the probability of the union of A and B.
• P(A B):
Probability that event A and event B both occur. This is also known as the probability of the intersection of A and B.

### What kinds of problems can the Probability Calculator handle?

The Probability Calculator computes an unknown probability, based on the value of related known probabilities. Here are the types of problems that the Probability Calculator can handle:

• Find P(A), given P(A').
• Find P(A), given P(B), P( B|A ) and P(A B).
• Find P(A), given P(B), P(A B), and P(A B).
• Find P(A), given P( B|A ) and P(A B).

• Find P(A'), given P(A).
• Find P(A'), given P(B), P( B|A ) and P(A B).
• Find P(A'), given P(B), P(A B), and P(A B).
• Find P(A'), given P( B|A ) and P(A B).

• Find P( B|A ), given P(A) or P(A'), P(B), and P(A B).
• Find P( B|A ), given P(A) or P(A'), and P(A B).

• Find P(A B), given P(A) or P(A'), P(B), and P( B|A ).
• Find P(A B), given P(A) or P(A'), P(B), and P(A B).

• Find P(A B), given P(A) or P(A'), P(B), and P(A B).
• Find P(A B), given P(A) or P(A'), and P( B|A).

If the above notation is confusing, see the frequently-asked question on notation, shown above.

### How can the Probability Calculator help me solve probability problems?

Solving a probability problem is a three-step process:

• Define the problem. Specify the research goal (what you want to know), based on the information you have.
• Analyze data. Apply the right analytical technique to achieve the research goal.
• Report results. Present the answer to the research goal.

The Probability Calculator provides a framework to help you define the problem. You select a research goal from the dropdown list box, and you enter known probabilities into one or more text boxes.

The Probability Calculator does the rest. It applies the right analytical technique to achieve the research goal. And it creates a summary report that describes the analysis and presents the research finding.

### What are the meanings of the various statistical terms used by the Probability Calculator?

To use the Probability Calculator and to understand the summary report it prepares, you need to understand some statistical jargon. If you encounter a term that you don't understand, visit the Statistics Glossary. All of the terms used by the Probability Calculator are defined in the glossary.

## Sample Problems

1. Bob is running in two races - a 100-yard dash and a 200-yard dash. The probability of winning the 100-yard dash is 0.25, and the probability of winning the 200-yard dash is 0.50. The probability of winning at least one race is 0.75. What is the probability that Bob will win both races?

Solution:

The first step is to define the problem. We begin by identifying the key events:

Let event A = Bob wins the 100-yard dash.
Let event B = Bob wins the 200-yard dash.

Then, we define the main goal, in terms of these events. For the main goal, we want to know the probability of the intersection of events A and B; that is, we want to know P(A B).

Next, we specify the known probabilities:

P(A) = 0.25.
P(B) = 0.5.
P(A B) = 0.75.

Now that the problem is defined, we enter the problem definition into the Probability Calculator. Specifically, we do the following:
• Identify the main goal: Find probability of intersection of A and B.
• Set the probability of event A = 0.25.
• Set the probability of event B = 0.5.
• Set the probability of the union of events A and B = 0.75.

Then, we hit the Calculate button. This produces a summary report that describes the analytical technique and computes the probability of the intersection of events A and B. Thus, we find that P(A B) = 0.00.
1. Mary is a successful pitcher for her college softball team. On average, she wins 75% of the time. However, when she gives up a home run, Mary wins only 50% of the time. She gives up a home run in half her games. In her next game, what is the probability that Mary will give up a home run and win?

Solution:

The first step is to define the problem. We begin by identifying the key events:

Let event A = Mary gives up a home run.
Let event B = Mary wins.

Then, we define the main goal, in terms of these events. For the main goal, we want to know the probability that both events occur; that is, we want to know the probability that Mary gives up a home run and Mary wins. This is the intersection of events A and B; that is, we want to know P(A B).

Next, we specify the known probabilities:

P(A) = 0.5, since Mary gives up a home run half the time.
P(B) = 0.75, since Mary wins 75% of the time.
P( B|A ) = 0.5, since Mary wins only half the time when she gives up a home run.

Now that the problem is defined, we enter the problem definition into the Probability Calculator. Specifically, we do the following:
• Identify the main goal: Find probability of intersection of A and B.
• Set the probability of event A = 0.5.
• Set the probability of event B = 0.75.
• Set the conditional probability B, given A = 0.5.

Then, we hit the Calculate button. This produces a summary report that describes the analytical technique and computes the probability of the intersection of events A and B. Thus, we find that P(A B) = 0.25.